Waste Management (WM) represents an important part of Smart Cities (SCs) with significant impact on modern societies. WM involves a set of processes ranging from waste collection to the recycling of the collected materials. The proliferation of sensors and actuators enable the new era of Internet of Things (IoT) that can be adopted in SCs and help in WM. Novel approaches that involve dynamic routing models combined with the IoT capabilities could provide solutions that outperform existing models. In this paper, we focus on a SC where a number of collection bins are located in different areas with sensors attached to them. We study a dynamic waste collection architecture, which is based on data retrieved by sensors. We pay special attention to the possibility of immediate WM service in high priority areas, e.g., schools or hospitals where, possibly, the presence of dangerous waste or the negative effects on human quality of living impose the need for immediate collection. This is very crucial when we focus on sensitive groups of citizens like pupils, elderly or people living close to areas where dangerous waste is rejected. We propose novel algorithms aiming at providing efficient and scalable solutions to the dynamic waste collection problem through the management of the trade-off between the immediate collection and its cost. We describe how the proposed system effectively responds to the demand as realized by sensor observations and alerts originated in high priority areas. Our aim is to minimize the time required for serving high priority areas while keeping the average expected performance at high level. Comprehensive simulations on top of the data retrieved by a SC validate the proposed algorithms on both quantitative and qualitative criteria which are adopted to analyze their strengths and weaknesses. We claim that, local authorities could choose the model that best matches their needs and resources of each city.